Season the beef with salt and pepper the evening before and refrigerate covered for the best flavor throughout this Korean Beef Chuck Roast. You can also shred the slow-cooked meat to make delicious Korean Beef Tacos.
Preheat oven to 350 degrees. Season beef with 1 tablespoon salt and 1/2 teaspoon pepper. Heat 2 tablespoons oil in a large 6-quart Dutch oven over medium-high. Add beef and brown on all sides, about 15 minutes. Pour off fat and wipe pot with a paper towel.
Reduce heat to medium and add remaining 2 tablespoons oil, onion, and garlic. Cook 3 minutes, stirring occasionally and scraping up brown bits on bottom of pan, until softened. Add ginger and cook, stirring, 1 minute. Add sake, mirin, broth, and gochujang. Bring to a boil, stirring to dissolve gochujang. Return beef and accumulated juices to pan and bring back to a boil. Cover and transfer to oven. Cook, flipping a few times and adding carrots halfway through, 2 hours. Remove pot from oven and stir mushrooms into sauce. Cover and cook until meat is fork-tender, about 30 minutes more.
Transfer beef to a cutting board and let rest 20 minutes. Meanwhile, remove vegetables to the board with the beef using a slotted spoon. Skim fat from sauce and bring to a boil. Cook until thickened like gravy, about 5 to 10 minutes more. Keep sauce warm until serving.
Remove string from beef; slice 1/4 inch thick. Serve with vegetables and sauce.